The disclosures in Korean Application No. 2000/34951, filed Jun. 23, 2000, and International Application No. PCT/KR01/01069, filed Jun. 22, 2001, are hereby incorporated by reference.
The present invention relates to a digital noise-shaping filter with real coefficients for delta-sigma data converters used in a digital amplifier, 1-bit digital/analog converter (DAC), 1-bit analog/digital converter (ADC) and the like, and a method for making the same.
The reasons for employing a conventional digital noise-shaping filter and problems with the digital noise-shaping filter are as follows.
1. Reasons for Employing Noise-Shaping Filter
1.1 Oversampling
Oversampling is typically used in a variety of technical fields such as a delta-sigma data converter, digital amplifier, etc. Oversampling means sampling of the original signal at a higher sampling rate than a normal sampling rate at which the original signal can be held and restored with no loss in information thereof. For example, assuming that an audio frequency band ranges from 20 Hz to 20 kHz, then sampling at the normal sampling rate signifies sampling of an audio signal at a minimum of about 40 kHz on the basis of the Nyquist sampling theorem. But, oversampling means sampling of the audio signal at a higher sampling rate than the minimum sampling rate based on the Nyquist sampling theorem. For example, 8-times (xc3x978) oversampling is defined as sampling the above audio signal at a frequency of eight times the minimum sampling frequency, or on the order of 320 kHz (see FIG. 1).
In case oversampling is carried out as mentioned above, the signal and quantization noise have spectra varying as shown in FIG. 2. In FIG. 2, the signal is indicated by a solid line and the quantization noise is indicated by a dotted line. The reason why the spectra of the signal and quantization noise varies as shown in FIG. 2 is disclosed in most teaching materials related to discrete signal processing (see: Alan V. Oppenheim, Ronald W. Schafer with John R. Buck, DISCRETE-TIME SIGNAL PROCESSING 2nd ed. pp 201-213 (Prentice Hall Signal Processing Series, Upper Saddle River, N.J., 1999)).
It can be seen from FIG. 2 that a band that the signal occupies on the standard frequency axis owing to oversampling is reduced in width compared to that prior to oversampling in inverse proportion to an oversampling ratio. An appropriate low pass filter can be used to remove a great portion of noise from the signal having such a reduced bandwidth. A noise-shaping filter can also be used to still further reduce the amount of noise energy of the band where the signal exists, by changing the shape of noise distribution by bands.
FIG. 3 shows spectra of the signal and quantization noise shaped by the noise-shaping filter, wherein the signal is indicated by a solid line, the quantization noise before being shaped is indicated by a dotted, straight line, and the quantization noise after being shaped is indicated by a dotted, curved line. From comparison between the quantization noise before being shaped and the quantization noise after being shaped in FIG. 3, it can be seen that the quantization noise of the signal band is significantly reduced in amount owing to the noise shaping.
1.2 Pulse Width Moldulation (PWM) and Requantization
PWM is one of methods for expressing a quantized signal (see FIG. 4). In this PWM technique, each discrete signal has a fixed amplitude (an amplitude on the vertical axis in FIG. 4), which represents a physical amount such as a voltage, and a pulse width on the time axis, which varies in proportion to the magnitude of the original signal. An appropriate low pass filter can be used to restore the resulting modulated signal with both the original signal and harmonic components into the original signal. The PWM technique is used mostly in a digital amplifier, delta-sigma converter, etc.
Signal modulation by the PWM technique necessitates a signal processor that has a higher degree of precision on the time axis than a sampling frequency of the original signal, in that the magnitude of the original signal is expressed not by a pulse amplitude, but by a pulse width. For example, for quantization of a signal sampled at 44.1 kHz into a 16-bit signal, the signal processor is required to have a processing speed of 44.1 kHzxc3x97216≈2.89 GHz. In some cases, a frequency that is twice as high as the above processing speed may be used for quantization according to a given PWM mode.
Further, the PWM cannot help generating undesired harmonic components due to its inherent characteristics. In this regard, oversampling must be carried out to reduce the harmonic components, resulting in the precision on the time axis becoming a higher frequency than that in the above example. For example, for 8-times oversampling, the signal processor is required to have an operating frequency of about 2.89 GHzxc3x978≈23.12 GHz.
However, it is practically impossible to embody such a high-speed signal processor. For this reason, the resolution of quantization must be lowered to a smaller value than the 16-bit value in the above example, which is typically called requantization. A requantized signal has a greater error compared to the original signal, which is expressed as noise components of the original signal. A noise-shaping filter is used to compensate for such an error.
2. Noise-Shaping Filter
A noise-shaping filter functions to shape the spectrum of quantization noise in a delta-sigma data converter. FIG. 5 shows the structure of a conventional noise-shaping filter. In this conventional noise-shaping filter, a digital input signal {circumflex over (x)} of b bits is quantized into an output signal {circumflex over (x)}+ens of bxe2x80x2 bits, where bxe2x80x2 is smaller than b. The component ens of the output signal is a noise component after the input signal is passed through the entire system. A component erq is a difference between a signal before being quantized and a signal after being quantized. The noise-shaping filter can shape the quantization noise by passing such a signal difference through an appropriate filter, feeding the resulting value back to the input signal and adding it to the input signal. A transfer function of the appropriate filter is defined as H(z).
The noise-shaping filter acts to reduce noise components at a specific frequency band in question by appropriately shaping the spectrum of quantization noise. A noise transfer function of the noise-shaping filter of FIG. 5 can be defined as in the following equation 1.                               N          ⁢                      xe2x80x83                    ⁢          T          ⁢                      xe2x80x83                    ⁢                      F            ⁡                          (              z              )                                      ≡                                            E                              n                ⁢                                  xe2x80x83                                ⁢                s                                      ⁡                          (              z              )                                                          E                              r                ⁢                                  xe2x80x83                                ⁢                q                                      ⁡                          (              z              )                                                          [                  Equation          ⁢                      xe2x80x83                    ⁢          1                ]            
where, Ens(z) and Erq(z) are z-transforms of ens and erq, respectively.
The noise transfer function can be derived from a conceptual diagram of FIG. 5 as in the below equation 2.
NTF(z)=H(z)xe2x88x921xe2x80x83xe2x80x83[Equation 2]
The noise transfer function exerts an important effect on the performance of a conventional noise-shaping filter. The conventional noise-shaping filter has a noise transfer function expressed by the following equation 3.
NTF(z)=xe2x88x92(1xe2x88x92zxe2x88x921)N xe2x80x83xe2x80x83[Equation 3]
where, N is a natural number, which is an order of the filter.
For example, a noise transfer function of a second-order (order-2) filter can be obtained as in the below equation 4 by expanding the noise transfer function of the above equation 3.
NTF(z)=xe2x88x921+2zxe2x88x921xe2x88x92zxe2x88x922 xe2x80x83xe2x80x83[Equation 4]
Similarly, a noise transfer function of a third-order filter can be expressed as in the following equation 5.
NTF(z)=xe2x88x921+3zxe2x88x921xe2x88x923zxe2x88x922+zxe2x88x923 xe2x80x83xe2x80x83[Equation 5]
The below table 1 shows coefficients of respective terms in noise transfer functions of second-order to seventh-order filters for 8-times oversampling, where all constant terms are xe2x88x921.
As seen from the above equation 4, equation 5 and table 1, the coefficients of the noise transfer function of the conventional noise-shaping filter are integers (see: J. M. Goldberg, M. B. Sandler, Noise Shaping and Pulse-Width Modulation for an All-digital Audio Power Amplifier, J. Audio Eng. Soc., vol. 39 pp. 449-460 (June 1991)).
The conventional noise-shaping filter is desirably simple to construct because the coefficients of the respective terms in the noise transfer function thereof are simple integers, but has a disadvantage in that it cannot maximize noise suppression performance.
Further, the higher order of the noise transfer function results in an instability in the entire system although it is able to improve the performance of the conventional noise-shaping filter (see: Alan V. Oppenheim, Ronald W. Schafer with John R. Buck, DISCRETE-TIME SIGNAL PROCESSING 2nd ed. pp 201-213 (Prentice Hall Signal Processing Series, Upper Saddle River, N.J., 1999)). That is, the order of the filter cannot unconditionally increase in consideration of the system stability.
Accordingly, there is a need for a noise-shaping filter to minimize an instability of the entire system and use a noise transfer function capable of maximizing noise suppression performance, more particularly coefficients of the noise transfer function, or coefficients having values other than integers.
Therefore, the present invention has been made in view of the above problems, and it is an object of the present invention to provide a digital noise-shaping filter and a method for making the same, wherein respective coefficients of a noise transfer function of the noise-shaping filter are not simple integers, but appropriate real coefficients.
It is another object of the present invention to provide a digital noise-shaping filter having real coefficients capable of maximizing noise suppression performance, and a method for obtaining the real coefficients of the noise-shaping filter.
It is a further object of the present invention to provide a digital noise-shaping filter with real coefficients which shows excellent noise suppression performance over those of conventional noise-shaping filters in the same order, and a method for making the same.
It is another object of the present invention to provide a digital noise-shaping filter with real coefficients which is operable in a lower order than those of conventional noise-shaping filters to obtain the same noise suppression performance as those of the conventional noise-shaping filters, resulting in an increase in system stability, and a method for making the same.
It is yet another object of the present invention to provide a digital noise-shaping filter which has real coefficients approximating optimum values so that it can have almost the same performance as the optimum performance without increasing a calculation complexity, and a method for making the same.
In accordance with one aspect of the present invention, the above and other objects can be accomplished by the provision of a digital noise-shaping filter for a delta-sigma data converter, comprising a noise transfer function expressed by NTF(z)=xe2x88x921+a1zxe2x88x921+a2zxe2x88x922+xcex9+aNzxe2x88x92N, the noise transfer function having real coefficients, where NTF(z) is a z-transform of the noise transfer function, a1, a2, . . . , aN are the real coefficients of the noise transfer function, and N is an order of the noise-shaping filter.
Preferably, the real coefficients of the noise transfer function may be obtained by 1) defining an objective function enabling a quantitative evaluation of the performance of the noise-shaping filter; 2) obtaining real coefficient conditions capable of optimizing the objective function; and 3) mathematically calculating optimum real coefficients satisfying the real coefficient conditions.
In accordance with another aspect of the present invention, there is provided a method for making a digital noise-shaping filter for a delta-sigma data converter, the digital noise-shaping filter comprising a noise transfer function expressed by NTF(z)=xe2x88x921+a1zxe2x88x921+a2zxe2x88x922+xcex9+aNzxe2x88x92N, the noise transfer function having real coefficients a1, a2, . . . , aN, wherein the real coefficients of the noise transfer function are obtained by the steps of a) defining an objective function enabling a quantitative evaluation of the performance of the noise-shaping filter; b) obtaining real coefficient conditions capable of optimizing the objective function; and c) mathematically calculating optimum real coefficients satisfying the real coefficient conditions.
Preferably, the real coefficients of the noise transfer function may be values approximating the optimum real coefficients, and the approximate real coefficients may be values approximated to four or more decimal places by a binary number.